YALE UNIVERSITY LIBRARY 3 9002 04972 3704 ART 3ag££ YALE UNIVERSITY ART LIBRARY ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE AGES , -*¦ BY K. WILLIS, M.A., F.R.S., &c. JACKSONIAN PROFESSOR IN THE UNIVERSITY OF CAMBRIDGE AND HON. MEMBER OF THE INSTITUTE OF BRITISH ARCHITECTS Reprinted from the TRANSACTIONS OF THE ROYAL INSTITUTE OF BRITISH ARCHITECTS ' Vol. I., Part 2, 1842 LONDON THE EOYAL INSTITUTE OF BEITISH AECHITECTS 9 CONDUIT STREET, REGENT STREET 1910 y\tt" 4S/~0 W CONTENTS PAGE INTRODUCTION ........ .1 SECTION I. ON THE GENERAL CONSTRUCTION OP THE VAULTS. 2 SECTION II. ON THE CURVATURE OF THE RIBS . '.» SECTION III. ON THE RIDGE RIBS, LIERNES, AND BOSSES . . 21 SECTION IV. ON FAN VAULTS . . 29 ADDITIONAL REMARKS TO SECTION II. ON THE CURVATURE OF THE RIBS . 41 ERRATUM .... .... . . .46 ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE ACES. BY E. WILLIS, M.A., F.E.S., &c. JACKSONIAN PROFESSOE IN THE UNIVERSITY OF CAMBRIDGE, AND HONORARY MEMBER OF THE INSTITUTE OF BRITISH ARCHITECTS. INTEODUCTION. IN the year 1568, Philibert de l'Orme published at Paris his treatise on Architecture. This is the first work in which the art of masonic projection is introduced, and it contains a very complete essay upon that subject, which our author denominates the art of describing " les traicts geomdtriques qui monstrent comme il fault tailler et cowpper les pierres," and which in fact is the. art of obtaining from the plans and designs of a proposed building the shape of the individual stones in a convenient form for the use of the mason. This first essay was followed by two works on the same subject by Maturin Jousse and Derand, which appeared nearly at the same time, namely, in the years 1642 and 1643. Other writers in France published upon this favourite subject, namely, Desargues in 1643, De la Eue in 1727, and Frezier in 1738. In fact, this has always been, and is now, a standard branch of architectural education in France, under the name of the " Coupe des pierres." It is joined with the corresponding art in carpentry, of which the first work was also written by Maturin Jousse, and published in 1627. In our own country this art has been rather neglected ; our only writers are Halfpenny, Art of Sound Build ing, 1725, and Nicholson, whose various works on this subject are too well known to professional men to need enumeration. Now although the merit of reducing this art to written laws and systems, and thus of giving it a place in the literature of science, belongs undoubtedly to De l'Orme and his immediate suc cessors already enumerated, it must not be supposed that he was the inventor of the art. On the contrary, the necessity for it plainly arose very early in the history of masonry. Indeed, De l'Orme never pretends to be the first inventor. His own language is that of a curious pedant in his art, and he manifestly introduces many new methods and forms ; but generally he talks of the matters which he is teaching as belonging to the ordinary practice of his age, which he is for the first time reducing to writing. It must be remembered, too, that the forms of the vaults which serve him for examples belong to the Italian style, and are very different from those of the Middle Ages ; so that granting to him the merit of applying and modifying the old Gothic methods to adapt them to these new forms, it is plain that much of originality is due to him. He, like all his contemporaries, imagined his own architecture to be a genuine restoration of the ancient classical styles ; and therefore he speaks of the Gothic vaults as " voutes modernes que les maistres magons ont accoustume" de faire aux eglises et logis des grands seigneurs " (p. 107) 2 WILLIS ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE AGES " Auiourd'huy ceux qui ont quelque cognoissance de la vraye architecture, ne suiuent plus ceste facon de voute, appellee entre les ouvriers La mode Francoise, laquelle veritablement je ne veux despriser, ains plustost confesser qu'on y a faict et pratique de fort bons traicts et difflciles." Frezier also, speaking of Gothic vaults, says with great truth, " Toutes ces naissances entre- lassees, et les intersections des moulures demandoient une grande intelHgence dans l'Art de la Coupe des pierres ; d'ou je conjecture, que c'est a rarchitecture Goihique que nous devons rap- porter l'origine, ou du moins l'adolescence de cet Art. Ma raison est qu'outre qu'il ne nous reste pas de Monumens antiques ou il ait ete mis en usage que pour des traits assez simples, c'est que dans l'enumeration que Vitruve fait des connoissances necessaires a un Architecte, il ne parle point de celle de la Coupe des Pierres ; en effet, la noble simplicite de I'Architecture des Anciens n'exercoit pas beaucoup le scavoir-faire des appareilleurs, qui n'avoient presque que des voutes cylindriques ou spheriques a conduire." (t. 1, p. xvii.) It becomes, therefore, a curious and interesting subject of inquiry to trace, from an examina tion of the structures themselves, what geometrical methods were really employed in setting out the work, and how the necessity for these methods gradually arose. Independently of the value of such investigations to the history of the science of construction, the knowledge of the methods actually employed would greatly assist us in the imitation of the works of each period. For the forms and proportions of every structure are so entirely dependent upon its construction and derived from it, that unless we thoroughly understand these constructions, and the methods and resources which governed and limited them, we shall never succeed in obtaining the master key to their principles, and instead of designing works in the style of any required age, we must content ourselves with merely copying them. The following paper must be considered as an attempt to sketch out an investigation of this kind, and in offering it to the body of practical men who are assembled in this Institute, I am not without hope that some of them may be induced to collect facts and examples by which this investigation may be carried on and completed. For it will appear, as we proceed, that most of the facts required are of such a nature that they can only be derived from the existing buildings by the aid of scaffolding, minute measurement, and close observation, which it is not often in the power of mere travelling observers to obtain. Now professional men are so commonly entrusted with the repairs or restoration of these old structures, that if they would take the opportunity of making the required observations in every case where scaffolds were erected about a building, and if such observations were trans mitted to the Institute, a few years would suffice to bring together a body of examples from which general rules might be deduced. It is only by comparing many examples that this can be done, for general rules deduced from single instances are commonly worthless. Next to complete buildings under repair, Euins afford the most valuable information upon construction. The best instructor of all, perhaps, is a building which is being pulled down, but such opportunities are always to be regretted. In ordinary cases, the upper surfaces of the vaults are so often covered with courses of rubble and concrete, rubbish and filth, and the lower surfaces with whitewash and paint, that when every facility has been obtained for examination, the jointing of the masonry and actual construction of the vault will still remain an unfathom able mystery. Section I. — On the General Construction of the Vaults. The ribbed vault of the Middle Ages, which is the subject of the present paper, differs entirely from the vaults of the Eomans. It consists, as is well known, of a framework of ribs or stone arches, upon which the real vaults or actual coverings of the apartment rest. These WILLIS ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE AGES 3 vaults are usually constructed of a lighter material and with rougher workmanship than the ribs upon which they rest, and between which they constitute, in fact, a kind of thin pannel. The construction of these ribs offers no difficulty in stone-cutting, each being, separately considered, a simple arch ; nevertheless the forms of its voussoirs can scarcely have been obtained without tracing on the ground the full-sized figure of this arch, from which to obtain the face moulds of the voussoirs. However, in the Norman ribbed vaults the arches are so roughly constructed that it may be supposed that a small portion of the arch only was described and a single face mould em ployed for the whole, and that the joints of the voussoirs were not divided upon a complete full- sized drawing of the arch. We shall see, however, that this full-sized drawing was very soon required in the early English period. Some of the Norman vaults are plain waggon vaults of rubble work as in the nave of the chapel at the Tower, or groined vaults of rubble as in the side aisles ; but it is unnecessary to dwell upon these very early specimens, in which no difficulties either of stone cutting or of curvature are yet introduced.* It is only where the groins of the vault are supported by ribs that the branching of these ribs in various directions from one abacus, and the different spans of the transverse and diagonal ribs, introduce difficulties, which were at first, as might be expected, somewhat clumsily botched over or evaded, and which afterwards led to the invention of the geometrical system, the rise and progress of which it is my purpose to trace. In Norman ribbed vaults, however, each rib springs independently from the abacus, and appears to have been erected without any reference to the management of the incumbent vaults. The latter are formed of rubble and irregular work, and may probably, in some cases, have been packed together without centering,")" or at least with a very rude framework ; and to accommodate the curvatures of the backs of the ribs, especially near the abacus where the ribs approach, these ribs are backed out where necessary with irregular stones and rubble work. There is an excellent example of this kind of vault in some apartments on the west side of the south transept of Peter borough Cathedral. These have been long employed as mere workshops, and the surface of the vaults being denuded of plaster, if it ever had any, its construction is plainly shown. The compartments are covered with a cross-ribbed vault, of which the diagonal arches are semi circular, and the transverse pointed. The span of the latter is about 13 feet. The voussoirs of the diagonal ribs are 12 inches square in the transverse section, and backed up, as already described, near the springing, so as to throw the spring of the vaults to a considerable height above the abacus. There are many other curious irregularities and peculiar forms in this vault, which is well worth examining. Another good example is in the castle at Newcastle-upon-Tyne, in a large vaulted apart ment about 27 feet by 20.J This has a central pillar, from which spring four transverse arches and four diagonal arches. Upon these the vaults rest. This is a very curious specimen, for every pair of these arches has a different span : that of the two transverse arches upon the long diameter of the room being 12 feet 3 inches each ; of the two shorter transverse arches, 9 feet ; and of the diagonal arches, about 15 feet 9 inches. All these arches are semicircular, and their crowns are all placed at the same level, to allow for which they are made to spring from different levels. Fig. 1 is a sketch of the springing of * Of plain Norman waggon vaults, there are two ex- the nave of the cathedral at the west end — onee probably cellent specimens at Norwich, one of them, under the the guests' hall, now the kitchen of a prebendal house. Bishop's Palace, of about twenty feet span, supported at f Vide an excellent paper by De Lassaux, on a mode of short intervals by subribs, plain and square edged, and erecting light vaults, Journal of Royal Institution, vol. i., resting upon semicircular plain corbels. A string cornice p. 224. of the usual Norman profile runs along the impost line of % Accurate plans and sections, with details of this vault the entire apartment, and is mitred round the corbels. and of the castle, are in the Vetusta Monumenta. There are The other specimen is an apartment on the south side of also some rough sketches in Carter's Ancient Architecture. WILLIS ON THE CONSTRUCTION OF THE VAULTS OF THE MfDDLE AGES FIG. J. these arches from the central pillar, c d is one of the small transverse arches, which having the least span, springs from the highest point c. ab is one of the greater transverse arches, springing from a below the level of c, and bf,ki,bh, are diagonal arches, whose span being the greatest, spring from still lower points, e, k, and g, so that the crowns of all these arches rise to the same level. With respect to the voussoirs, it will be seen that although the backs or extrados of the voussoirs which constitute the arches are concentric to the soffits of these arches, yet these are backed up by a rude wall of rubble work, as at m, n and p, upon which the rough vaults are supported, and this is necessary to get rid of the difficulty of resting a vaulting surface upon arches whose cur vature and level at the springing are so different. Fig. 2 is another example, which perhaps shows the nature of this difficulty more clearly. This is a portion of a vault from the ruins of Finchale Priory, in the county of Durham, a b c d are two ribs or arches which spring with very different curvatures from a single pillar. The vault is in this case formed of long thin stones, packed one upon the other, as shown at e p g. And as the different curvatures of the arches throw the surface at / much higher than that at e, the difference of level is made up by courses of rubble upon the lowest extrados of the two only between e and m, which are gradually thinned off at the latter point, and enable the stones of the vault to be laid straight from one arch to the other. From these and many other examples it appears that in this early stage of rib vaulting' the arches or ribs consist of independent and separate voussoirs down to the level course from which they spring, as in fig. 2, where a and c, a and c, are separate stones roughly jointed at the back, instead of being each got out of a single stone, as in the subsequent structures. Also the back or extrados of these ribs is con centric with the soffit, and thus formed without reference to the arrange ment of the incumbent vault, which was plainly a subsequent and separate consideration ; the ribs being, after their erection, backed out at the springing so as to accommodate the curvature of their extrados to the reception of the vault. The vaults of the choir at Canterbury have in like manner their ribs formed of separate stones, from the abacus upwards, and backed up where necessary ; the voussoirs of the ribs are also very small and numerous : I counted about a hundred in one transverse rib of the north eastern transept in a span of 30 feet. These ribs are very richly moulded, but the workman ship is exceedingly rude. To this rough construction of the spandrel succeeds all at once a more artificial structure, bespeaking a great advance in the art of masonry ; and it is remarkable that this new construc tion once introduced remains with very slight change to the very latest period of rib vaulting. Fig. 3 is a diagram to illustrate this construction, and is principally derived from the south transept of Westminster Abbey, but is not drawn to scale. The left hand half of this diagram represents a portion of the vault in perspective, including the entire spandrel solid (if I may be allowed the expression), which is contained by the two semi-diagonal ribs a d a e and the wall. a b c is the transverse rib, and on the right hand of the diagram the vertical section through this rib is exhibited, g h is the string moulding upon which the clerestory windows rest, and the arch which contains these windows springs from l k at a considerable height above the springing, a, of the vault ribs. This is a very universal arrangement of clerestory vaults, and FIG. 2 . WILLIS ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE AGES is productive of great beauty and convenience, but it leads to some difficulty in the form and arrangement of the vaulting surface a k p d, for as this is contained between two arches or ribs, a d k p, which spring from different levels, it follows that this surface must be skewed back at k in a very peculiar manner. This is shown by the perspective. The junction of the solid mass, a k q, with the clerestory wall is therefore bounded by parallel vertical lines, one of which is a k, and this mass is always built of solid masonry bonded into the wall and forming a part of it.* It is from the level of k q that the real rib and pannel f construction of the vault begins, for separate ribs are erected upon the surface of this solid, and connected by vaults of a light material. From below, however, if the vault be painted and deco rated, this change of construction at k q is disguised, or, in other words, the decorative construction of the vault exhibits the rib and pannel from the abacus a upwards, but the mechanical construction is of solid masonry from a to q, and of rib and pannel work only above this level. The point q of this change of construction is commonly at about half the vertical height of the arch, as shown in the drawing, but is not necessarily at the same level as the impost k of the clerestory rib k p. The peculiar construction of the solid mass a k q is better shown in the section at c m n. The ribs of the vault converging downwards to c, their mouldings become entangled as it were, in a manner that will be subsequently explained. At a point m, about halfway up the solid, they are, however, freed from each other, and separated by the divergence of the ribs. Now, between c and m the solid is built of horizontal courses of masonry, generally each of a single stone, and its level beds cut the curved mouldings obliquely in front. Above m the ribs are each built separately of voussoirs, having their beds properly inclined to meet the axis of curva ture p of the rib, and these ribs are backed and united by solid masonry which connects them with the wall, and which appearing between the ribs seems to be a portion of the light vaulting surface, such as is really employed higher up. From the upper surface n of the solid, each rib n b is still built as from m to n with voussoirs, but upon these ribs rests the light thin vault or pannel, shown in section on the right, and in perspective on the left of the diagram. It is remarkable that the courses of the vaults are not laid level, but are in most cases made to incline downwards upon the diagonal rib. Thus in fig. 3 the ridge p d is level, and also the ridge dbs; but the courses of the vaults incline considerably downwards from f k and from q b towards the diagonal rib a d. These courses, in Westminster transepts, are of a light-coloured stone, probably chalk, interrupted, at regular intervals, by a course of a darker stone ; and the ridge p d, which has no rib, is also formed entirely of this darker stone, laid in the serrated manner shown by the drawing. The * This block of masonry appears to have been termed the tas de charge, for Philibert de l'Orme defines this term thus : — " Ce sont les premieres pierres que on voit sur les angles, et monstrent le commencement et la naissance des branches, des ogives, tiercercms, formerets et arcs doubleaux." These five terms are the names of the ribs, as we shall see presently. f I employ the term rib and pannel work to distinguish that mechanical construction of the vaults of the Middle Ages, in which a frame work of ribs is made to support thin superincumbent vaults in the manner of pannels, from the vaults of solid masonry which were subsequently introduced, in which the stones are closely jointed through out, and the ribs and pannels merely carved on their lower decorative surfaces. This latter system, which is in fact derived from the original Roman or earlier vaults, is wholly adopted in Henry the Seventh's Chapel at Westminster. But the two methods are, as will appear in the course of this paper, mixed together in the greater number of examples. B 6 WILLIS ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE AGES dark courses are rather broader than the light ones, and there are four or five courses of the light between each of the dark. The surface p k d is also slightly concave or domical, and may therefore have been laid without any centering, since each course would support itself.* These peculiarities may all be found with some variations in other vaults of the same age. What might have been the reason for this downward inclination of the courses it is not easy to say, but it is very common, especially in the earlier examples. Some have supposed it to have arisen from the courses having been laid to meet the bounding ribs a k p a d at respec tively equal distances from their springing, which would certainly produce the effect in question, since the diagonal rib is so much longer than the others, but the downward inclination is greater than that which would arise from this cause. In some examples the slope seems to be derived from the courses having been laid so as to meet the diagonal rib at right angles. The perspective effect which arises from the arrangement is curious, for the vaulting surfaces a d b a b e are really very nearly coincident with a single surface extended from a d to a e, or in other words, a horizontal rod passed upwards along the backs of the ribs a d a e would very nearly touch the two vaulting surfaces and the back of the rib ab, But the effect of the inclination of the courses is to make the rib a b appear in perspective as if prominent downwards from such a surface, and consequently gives to the entire solid spandrel fdsa the effect of a kind of fan vault or vault with a polygonal horizontal section. Above the vaults are commonly laid a thick irregular course of rubble work, which again is also often covered with a coat of a kind of concrete. The vaults of Westminster Abbey, with the exception of the western compartments, those of Exeter, Winchester, Hereford (with the exception of the south transept and tower), Wells, Ely, Kedcliff Church, Bristol Cathedral, and many others, are thus covered. These upper coverings appear to have been abandoned in the later periods, but not universally. Those of the western compartments of Ely choir seem to have been subsequently picked off, perhaps by Essex, to lighten the vault. But the vaults of the western compartments of Westminster, and of the south transept and tower of Hereford, are left bare on the upper surface, and these vaults, instead of being built with small brick-like stones, are composed of long thin slabs. Also the ribs themselves are, in some later examples, formed of a few long bar-shaped voussoirs instead of the small and numerous pieces of the earlier examples. Thus in the transept of Westminster, n b consists of thirteen or fourteen stones, but at the west end of the nave, of six only. The employment of the solid mass of masonry aiq enables the ribs to approach more closely at the springing, and also reduces the actual span of the vault, for q n is the real span of the vault, instead of a c, which is the apparent or decorative span. Thus about one-sixth of the span is saved. The early ribs are formed as in fig. 4 (from St. Saviour's Church, Southwark), the vaulting surface resting only on their backs ; but the later ribs are rebated for the reception of the vault ing surface, as shown in fig. 14, by which greater depth and strength is given to them without necessarily increasing their projection from the surface of the vault. To return to the proper subject of this paper, namely, the evidences of a geometrical method in setting out the work : The solid spandrel from c to m, fig. 3, has been said to consist of level courses of masonry, and to contain all that portion fig. 4. of the vault in which the mouldings are entangled and partly concealed by the approximation of the ribs. Now two ways may be conceived in which these mouldings may have been worked out of the stone. The spandrel may have been built solid, and after the ribs m n were set up, the mouldings of the entangled ribs from m to c may have * Vide the essay of De Lassaux. WILLIS ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE AGES ~l\ FIG. 6. been worked gradually downwards, first in block, and afterwards in detail, and in this way the various interlacings and interpenetrations of the mouldings would develope themselves as the workmen proceeded. The other way would be to project geometrically upon each bed of the stones which con stitute the spandrel from c to m the mouldings of each rib in its proper position, and thus would be shown which mouldings remained prominent and which were covered by others ; and that this method was employed I shall show from examples. Of course a perfect existing building affords no opportunity of examining the beds of its stones, but if the beds of a newly pulled down structure be examined by carefully scaling off the mortar, the mason's lines will be found to remain as freshly upon the surfaces as when they were first set up, and from these it will be seen that a complete geometrical method was used to obtain the intersections of the mouldings. jjpf* Fig. 6 is a plan of one of the spandrel stones, taken down from the side aisle vault of St. Saviour's Church, Southwark, in the course of its demolition, in 1839. This drawing is reduced to one-eighth of the original, and the lines and marks which it exhibits are carefully copied from those that were found upon the surface or bed. Upon this were traced lines parallel to the direc tion of the wall and of the several ribs of the vault respectively, as p g in the direction of the wall, a b for the transverse rib, a c and d e for the diagonal ribs. The vault to which this spandrel belonged was of an irregular plan, four- sided, but having each side of a different length, to accommodate which, one of the diagonal ribs d e is made to spring from a point a little removed from the intersection a of the other a c with the transverse rib ab. Upon each of these lines the profile, or as much as is required of it, of each rib is traced, evidently by means of a templet, that is, of a pattern of the rib cut out in some thin material, which was held down in its proper place upon the stone while its outline was scratched or cut by carrying some sharp-pointed instrument round its edge. Thus the first traced was manifestly the complete profile abBefcdol the transverse rib. Next, on the left hand side is the profile b cgk of the diagonal rib, and this by its projection obscures and renders unnecessary a portion & a of the former rib ; and that this profile was traced subsequently to the former one is plain by the omission of all that is covered by it to the right of b. The wall-rib g f was next traced, and this obscures the portion g k of the former rib. In like manner, on the right hand side, the profile c e was traced, obscuring the piece c d of the first rib. At g the stone is broken. The mouldings that were thus shown to project appear to have been then completely worked down to the outline thus obtained, excepting at one or two places, such as at e and/, where the trace of the templet still remains within the real edge of the bed. The compartments thus vaulted in St. Saviour's side aisles were not rectangles but irregular trapeziums, and the diagonal ribs were slightly twisted upon the plan, as shown with some exaggeration in fig. 8. I have seen this in some other examples, but whether it results from design, with a view to dispose the branching ribs to better advantage, or from bad workmanship in not setting out the ribs in the tas de charge or solid block at the proper angles, and therefore making it necessary to warp their direc tions to enable them to meet at the crown, I am unable to decide. To draw these figures upon the beds of the stones, it is necessary to know in each case the quantity of projection of every rib, or, in other words, the points c b e to which the ribs extend FIG. 8 8 WILLIS ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE AGES at each joint, and against which the front of the pattern or templet must be placed before its outline can be traced. These are, however, so easily and obviously obtained by drawing on the full scale the elevation of each rib in its own plane, with the joints inserted after the manner of c m, fig. 3, that we may assume that this was the method employed. A diagram of this kind is to be found both on the upper and lower bed of each stone, and the lines a, d, &c, which extend to the outsides of the stones, are scored vertically down the back for the purpose of making the two diagrams coincide. I may add, that every stone which I examined that had appertained to the spandrels exhibited similar lines and traces of mouldings ; and as the pulling down of the buildings was proceeding during my examination, I had the opportunity of seeing the stones in their fresh state, and of myself scaling off the mortar. The mouldings of these specimens are sufficient to show that they belong to an early period of the early English style. Fig. 7 is an example of a similar kind belonging to the Perpendicular period, and taken from one of the spandrels of a complex vault which formerly covered the extreme north-western compartment of the nave of Canterbury Cathedral, the lower story of the so-called Lanfranc's tower, which becoming ruinous, was taken down a few years since, and is now replaced by a modern copy of the south-western tower. The stones of the original vault, however, were carefully deposited in the nave of the cathedral and in the yard at the time of my visit, and I found the surfaces of these spandrel stones covered by lines and profiles of mouldings similar to those already described from St. Saviour's, showing that the same method had continued in use from the period of its first introduction. The number of ribs that spring from each angle of this vault being seven, including the two wall ribs, made it necessary to employ two stones in each of its upper courses at least, and accordingly the bed represented in fig. 7, being only a portion of the entire spandrel, contains but four of these ribs. I have selected this one out of many others that I copied, because it is evident that a set of lines first drawn upon it were rejected because the stone was not large enough to contain the ribs, and another set was subsequently drawn and actually em ployed, which gives a somewhat additional interest to this example. a b a G are the rejected lines, and d a por tion of a profile of a rib belonging to a g. b d be b f b g are the true lines, drawn each parallel " ' ' - ' to its own rib, as shown by the plan of the vault. The profile d m appears to have been the first drawn, then n e p, then q f, and so on. The marks at s, t, i, &c, are apparently for the purpose of distinguishing the true lines from the false ones. The average thickness of the courses in this spandrel is about ten inches. As the beds of the stones are horizontal in all these cases, the upper beds necessarily cut the mouldings of the ribs at an acute angle or feather edge, and this sometimes occasions the edges of the stone to fly off. On the uppermost surface of these stones (as at m, fig. 3) every rib has its own slant bed provided and worked square to the plane of each elevation, so as to give a firm footing to the separate ribs that all start from this upper stone. One consequence of thus allowing, the bed of the stone to cut the arch line obliquely is, that the templet or pattern from which the mouldings are traced upon, the bed becomes too short WILLIS ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE AGES 9 in the oblique section ; for (fig. 5) if m n s p t be the upper stone, m n the bed from which the detached ribs start, and if m n be the transverse depth of the mouldings, then it is evident that on the lower bed p t the depth of the same mouldings p r will be considerably greater on account of the obliquity of the section. Now in the examples that I have examined, I found that the same templet had been employed to trace the mouldings upon m n, upon p r, and upon the other oblique beds below, and consequently these mouldings were drawn in and contracted very disagreeably, as shown by the dotted line nq, p q being taken equal to m n. This is a curious proof of the rough neglect of minute circumstances by the Gothic masons of which plenty of other examples might be adduced, and it must be confessed that the false lines so produced are not perceptible from below. I believe we might in many cases reduce very considerably the expenses of our construc tions if we had courage to imitate our ancestors in this respect. In the Canterbury example the angle m p q=110°, and the depth m n of the mouldings is 5 inches, consequently it may be easily calculated that p q is half an inch too short ; and the effect of drawing the point q half an inch out of its true place compared with ten inches, which is the thickness s t of the stone, is very perceptible when the stone is looked at upon the ground, but when in its place aloft I have no doubt this error was perfectly unappreciable. FIG. 5. Section II. — On the Curvature op the Eibs. The next set of examples which show the necessity of a geometrical system will be found in the construction of the intermediate ribs of vaults, and in the management of the curvature of the ribs generally. The limits of this paper will not allow me to enter fully into the descrip tion of the different classes of vaults in the decorative sense, neither is it necessary for my present purpose ; I shall therefore briefly state the different steps by which they appear to have been led on from the simple cross-ribbed vault to the fan tracery. The plain cross vault, Eoman in arrangement, but pointed and with the addition of ribs upon the groins, is to be found at Salis bury, Gloucester nave, Canterbury choir, Wells nave, Beverley, Westminster choir, and in all the French cathedrals. Simple intermediate ribs were first added between the wall ribs and diagonal ribs, and between the transverse ribs and diagonal ribs. Thus, in fig. 9, a vault with intermediate ribs is represented in a diagram upon the principle of delineation, which was first employed by Mr. Ware, in his admirable treatise on this sub ject.* a b k l are the points whence the ribs spring, and between which is given the plan of the vault in a kind of diagonal per spective, a c, b c, k c, l c are the diagonal ribs or great cross springers (termed croisee d'ogives by De l'Orme). a e a/ a g are the intermediate ribs of that spandrel of the vault which lies nearest to the eye in the diagram. These intermediate ribs are termed the tiercerons by De l'Orme, which being a very con venient word, I shall employ. Now in clerestory vaults, the transverse dimension of every compartment is commonly about double that of its longitudinal dimension, a d will, therefore, be the transverse rib of the vault, and a e the rib which lies next the wall of the clerestory or the wall rib, as I have ventured to call it, the formeret of De l'Orme. fig. s. * Tracts on Vaults and Bridges. 10 WILLIS ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE AGES In this figure I have shown one tierceron between the transverse and diagonal ribs, and two between the wall rib and diagonal rib ; but the number varies in different examples. The figure agrees with the vaults of the choir of Lichfield and the south transept of Hereford. One tierceron in each space dca and c a e is to be found in Lichfield nave and Lady Chapel, Norwich cloister, Exeter side aisles, Lincoln nave, Westminster nave and cloisters, and in the vault at the intersection of the nave and transept at Amiens Cathedral. Sometimes three and one are employed, as in Exeter nave, or three and two, as in Norwich nave. Much of the effect and character of these vaults depends upon the curvature of these tiercerons, and also upon that of the diagonal and transverse ribs between which they are placed ; and even in the simpler vaults, which have only the diagonal and transverse ribs, this curvature governs the character of the vault by determining the form of the spandrel solid. It is evident, indeed, that if a given parallelogram is to be vaulted with a groined and ribbed vault, and the crowns of the arches are all to be nearly at the same height, that some geometrical difficulties will be introduced in the management of the forms of arches of such different spans as the transverse, longitudinal, and diagonal ribs, and this difficulty was much greater before -the pointed arch was introduced. The vault of the castle at Newcastle, already cited and explained, is a parallelogram of 27 feet by 20, and the three spans are here accommodated by employing semicircular arches and stilting them at their imposts to bring their crowns to the same level. This, as I have shown elsewhere,* was the Eoman expedient, and was employed in the baths of Dio cletian and Caracalla, of course without the diagonal rib. The side aisles of the chapel in the White Tower of London, however, have plain groined vaults without ribs upon a parallelogram, exactly upon the same principle as that of Diocletian's baths, and with the waving groin, which I have, in the passage just referred to, shown to be a necessary result of such an arrangement. The side aisles of the nave of Peterborough Cathedral are examples of the same difficulty, which is overcome without the use of pointed arches by the new expedient of employing a seg ment less than a semicircle for the diagonal rib, or, in other words, by placing the centre of the circle below instead of above the level of the impost or springing. The parallelogram to be vaulted in this case is about 15 feet by 18 feet, but from the massive construction of the piers the bounding ribs are contracted in their spans, especially that on the long side which is next to the nave. The spans of the transverse and longitudinal ribs of the vault are 12 and 15 feet, and of the diagonal rib 21, and their crowns are nearly at the same level. To meet this difference of proportion between the spans and height, the longitudinal rib is slightly stilted, and the transverse rib very much so ; but the diagonal rib is a small segment of a circle, and therefore springs off the abacus at a considerable angle, and in a manner totally at variance with the two neighbouring ribs, which rise from their imposts with a slight inclination back wards, forming horseshoe arches. After pointed arches were introduced, the difficulty of adjusting these three arches was greatly diminished ; but notwithstanding the possibility of making pointed arches of any pro portion of height and span with their centres of curvature upon the impost line, it will be found that the old expedient of placing the centres of curvature above or below the impost line, for the better adjustment of these arches, was still retained in pointed architecture until the four- centred arch was brought in. I shall proceed to examine this more at length, since it involves so much of the characteristic appearance of these vaults. The ribs in early specimens consist each of an arc of a single circle, but in later examples of two arcs of different radii conjoined so as to form the half of what are termed four-centred * Architecture of the Middle Ages, p. 72. WILLIS ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE AGES 11 arches, since the term rib is applied to half the arch. Some ribs are, even in complex vaults, formed of three arcs, as I shall presently show. In the first place, however, I will speak of ribs of a single arc only. Eibbed vaults may have horizontal ridges, or they may be domical ; that is, the point e of the ridge (fig. 9) may be at the same level or may be lower than c, and it may be also curved from c to e, in which case, as far as I know, it is always either straight or concave on its lower side, but not convex.* Sometimes, to suit particular cases, as, for example, to admit of a high window, the apex e of the wall rib is thrown higher than c. Domical vaults are much more common on the Continent than in England, and especially run into excess in the middle age vaults of Italy ; however, we have some specimens of them, as, for example, in the nave of Worcester. The form of the ridge is, however, one of the first things that should be observed in examining an existing vault, and is also one of the first things that must be settled in a proposed vault ; for the form of the ridge decides the relative altitudes of the summits or crowns of the ribs. Now supposing a rib to consist of a single arc of a circle, we may either place the centre of this arc upon the impost level, or we may allow it to be placed above or below that line. But the plan of the vault gives the span of each rib ; also, when the height of the vault and the form of the ridges are determined, the altitude of the crown of each rib is also given. If, therefore, the centres are to be upon the impost line, the radius of each rib is given by these conditions ; but if the centre may be above or below, we may for each rib take any radius we please. For let a b (fig. 9a) be the span of a rib which is given by the plan, and b c the height of its crown, which is also given, as already explained. Then since the rib is an arc of a circle, which must pass through the two points a and c, its centre must be on the line e d, which is a perpendicular upon the middle point of the chord A c. If, therefore, the centre of the arc is to be on the impost line a b, it can only be at d, where the two lines e d a b intersect ; but if we are allowed to place it above or below this impost line, it may be at any points d or e upon the line e d. One set of workmen appear to have confined themselves to the first practice, and another set to have allowed themselves the second ; and it would be a very desirable thing to ascertain the exact curvatures of the ribs of a great many of these vaults, at the same time noting also their general character and appearance, with a view to determine the practice of the different schools, as well as to study the results with a view to improve modern practice, which is perhaps more deficient in this matter of curvature than in any other. In the above proposition I have supposed that the form of the ridge was settled, or at least the altitudes of the crowns of the ribs, before the radii were determined, but we may suppose various other methods of proceeding ; for example, that the pointed arches of the ribs should all be similar, that is, that they should all have the same ratio between their span and altitude. This rule produces a highly domical vault, because the crown of the diagonal rib is necessarily thrown very high above that of the other two. The Italian mediaeval vaults appear to be governed by this rule as nearly as the eye can detect. But in England the ridges of the vaults are most commonly level. In some specimens, however, other principles govern the relation of the curvatures ; for example, in plate 77, Pugin's Specimens, is a diagram to show the curvature of a vault on the east side of the cloisters at Westminster. In this vault the ribs are each a single arc of a circle, with the centre upon the impost line, and the diagonal rib has the same radius as the transverse * The ridges of fan vaults are convex downwards, but they belong to the class of vaults with four-centred ribs, 12 WILLIS ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE AGES ribs* so that the vault is highly domical in its structure ; it has no ridge ribs, but the ridge necessarily rises to the centre of the vault, since the crown of the diagonal rib is by this con struction thrown so much higher than that of the transverse ribs. This principle of employing a common radius for the diagonal and transverse ribs agrees with many other examples, as I shall presently show. On the other hand, there is also in Pugin's Specimens (vol. ii. p. 29) an excellent detailed drawing of the vault of the Lady Chapel, Southwark, in which the ridges are horizontal. The compartment is 20 feet 6 inches square, and the spans of the bounding arches are 14 feet 6, and of the diagonal arch 20 feet 8 inches. The ribs or semi-arches are each formed of a single arc of a circle struck from a point 8 inches below the impost level, and with a radius of 9 feet 5 inches. The entire diagonal arch is struck from a single centre 2 feet 7 inches below the impost level with a radius of 11 feet 3 inches, and is therefore a segment of a circle instead of being a pointed arch. In many examples I have seen that the transverse rib is struck from centres above the impost level, and therefore its arch is a horseshoe, and in the same vault the diagonal rib will often have its centres below the impost level. On the contrary, in the north transept of Hereford Cathedral the centres of all the vault ribs are very considerably below the impost level.f I have shown that when the centres are allowed to be placed out of the impost level, there is a choice of many radii for the curve of a rib of given span and altitude, and of course as the radius in creases, the rib approaches nearer to a straight line. But the effect upon the general form of the spandrel solid fdba, fig. 3, or d c e a, fig. 9, is the principal point to be attended to, and this is best appreciated by considering the form which its middle plan assumes, that is, the form of a horizontal section taken at about halfway up the arches, as at k q, fig. 3, or perhaps a little higher, as at m n p, fig. 9. If the ridge ribs are level, as in this figure, the plan at the crown of the vault d c e will necessarily be a rectangle, but by different arrangements of the curvature of the ribs we may make the middle plan mnp assume any figure we please. Thus in the simple vault with transverse and diagonal ribs only, as in fig. 3, we may, by making a b more or less curved, cause it either to lie between a d and a e in such a manner that a horizontal rod which touches these two ribs halfway up, will also touch a b ; or we may make the middle point of a b lie in front of such a rod, the effect of which is to make the spandrel solid appear convex in front, and approach to the form of a fan- vault, which is the case in Pugin's second example above quoted ; or else we may make the middle point of a b lie in the opposite direction, so as to make the solid concave in front : of each of which dispositions examples may be found. Similarly, when intermediate ribs are added, as in fig. 9, they may be disposed in various ways so as to affect in a similar manner the form of the middle plan, and throw it into various figures. By going to the upper surface of a vault, and looking down into the pockets or cavities which lie over the shafts, and which are in fact the insides of the spandrel solids, the middle plan will be more distinctly seen than from below, and in making architectural notes its general form at least should always be recorded. Since this middle plan affects the character and appearance of the vault so materially, * The arches in this example are very highly pointed ; 7 feet 8 inches ; the span is 11 feet 6 inches (so that the the compartment is 16 feet by 18 feet 2 inches, and the height is two-thirds of the span). The versed sine, or dis- height of the crown of the vault 17 feet 7 inches, or about tance of the middle point e of the chord a c from the arch, 18 feet if measured to the vaulting surface. The height is 7| inches. The radius obtained from these measures, is, therefore, the same as that of the long side of the either by calculation or by laying down the arch to scale, parallelogram, which perhaps was the principle that in comes out 18 feet 5 inches, and it also appears that the this case determined the proportions. vertical distance of the centre of the segment a c below the t These arches of Hereford north transept are so exees- impost line is very nearly equal to the altitude e c of the sively straight-sided that I measured them carefully, and crown of the arch above that line, with the following results. The height b c (fig. 9a) is WILLIS ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE AGES 13 I will show how, by a very simple construction, it may be employed in giving any desired form to the spandrel solid ; and although I do not mean to say that it was so employed by the Middle Age architects, it may be made useful in examining and comparing their works. Let fig. 9 (page 9) be a vault of which the plan is given, and of which the height of the apices of each rib, e, g, f, c, e, d, are also determined, as well as the middle plan p, v, s, n, r, m. The ribs being each a single arc of a circle, it is required to find the radius and centre of each rib. Now we have given for each rib three points through which it must pass : namely, the springing at a, the middle point, and the apex ; therefore the question reduces itself to the common workman's problem, Given three points to describe an arc of a circle through them.* The complete construction for a vault with tiercerons is shown in fig. 10. Let a b c d be the plan of the vault, a e the diagonal rib, a f ag the tiercerons, a h the clerestory rib. Draw l m of the required figure for the transverse rib l k, and also z p q the figure of the clerestory rib, which (as is shown at page 5) will be raised upon stilts, as at a p. Find the point s at half the height k m of the transverse rib, and draw r s perpendicular to a k. Draw the middle plan r w z of the spandrel according to the required form, also l z the plan of the ribs upon the abacus. To find, then, the curvature of any rib, as for example of the tierceron a p. If the ridge rib e k be horizontal, the height of the point f will equal that of the point k. Set off f/ per pendicular to a f and equal to k m ; and because r is the seat of the perpendicular from the point where the tierceron passes through the middle plan, set up r t perpendicular to a p and equal to r s. Then will l, t and / be three points through which the required rib passes, and the centre h found by the common rule, will be the centre of curvature of the rib. In the same manner will be obtained the centre g of the diagonal rib l m e. The arrangement of the rib upon a g is more difficult, for as it lies between two ribs, a h and a e, one of which is stilted above the other by a distance z p, it is necessary to stilt also the rib upon a g by about half that distance. Set up, therefore, a perpendicular t fc equal to about half z p, and obtain the other two points b and d as before, z will be the centre of curvature of k b d. _ Any other forms may be given to the middle plan, as shown at the angle c of the same figure by x y z, or at x w z, or at b and l in fig. 9. I have drawn the diagram fig. 10 upon the hypothesis that the middle plan shall be taken exactly at half the vertical height of the arch, and that the ridges shall be level. If, however, the ridges be curved into any given figure, that FIG. lO. * According to Philibert de l'Orme, the simple geometri cal construction by which an arc of a circle is described through three given points (not in one straight line), is strangely termed " Maniere de trouver les trois poinctz perdus." The curved templet or wooden sweep which is obtained from this process is called the " cherche ralongie.'' — Inventions pour Men bastir, p. 21. The fact of this problem having a familiar name shows it to have been in common use amongst workmen. Our workmen similarly term the mode of setting out a rectangle (from Euc. i. 47) the rule of six, eight, and ten, these being the convenient whole numbers of feet that possess the required property, namely, that the sum of the squares of the two first shall equal the square of the last, or 36 + 64 = 100. This rule is given in Vitruvius (b. ix. c. 2) from Pythagoras, and the numbers of feet used are three, four, and five. 14 WILLIS ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE AGES curvature will give the separate altitudes of the crowns of each rib, and therefore, instead of setting up k m f/ e e g d all equal, they must be made respectively equal to the altitudes thus obtained from the given ridges. Also it may be better to take the middle plan higher up, because the form of the spandrel will be developed more clearly ; for example, let it be taken at a point midway between the springing and the crown of the arch measured upon the arch itself. The point s being assumed halfway between l and m, the remainder of the process will be the same, but the middle plan nw z will come out larger in proportion to the rest of the figure. Again, I have drawn the transverse rib first, but it may be better to begin by drawing the diagonal rib a e first, as that, from its greater span, is the most difficult to manage ; in which case the point m must be assumed at the proper height for the middle plan, which will then be derived from the perpendicular m w instead of from the point s and perpendicular s r. I have reason to believe that the diagonal rib was often the first rib settled by the mediaeval architects, as it so often appears in the form of a segmental or semicircular arch. The want of its point is concealed by the boss. A common method of executing modern Gothic vaulting, especially in plaster, has been to obtain the form of the diagonal ribs, as a c, fig. 9, by ordinates from those of the transverse ribs, ad, a e. The effect of this is, that the entire vault appears as if it consisted of two waggon vaults, crossing at right angles, and a horizontal rod moved from top to bottom in contact with any pair of opposite arches b m a e will touch the diagonal ribs b c a c and the connecting vault ing surfaces throughout. This is the genuine principle of the Boman and Italian groined vault, but is altogether foreign to the principles of Gothic architecture, in which every rib should spring as a separate and independent arch, and in which the elliptic curves produced by this method are totally at variance with the characteristic forms of the style. De l'Orme first taught this method of projecting diagonal groins by ordinates, and applied it to the wooden cradling of an Italian vault in his Nouvelles Inventions pour bien bastir (1578, p. 20). This form of vault, however, is a kind of square dome (cul defour quarree), but requires the same diagonal curves for its groins (or rather ridges) as the Eoman cross vault. The latter vault is not amongst the examples of stone-cutting in De l'Orme's book. The square dome is also given by Maturin Jousse with the same method of ordinates (Le Secret d'. Architecture, 1642, p. 127), who also gives, for the first time, the Eoman groined vault (Voute diAreste quarree, p. 157) under different shapes and plans, which becomes a standard example in all the succeeding writers upon the subject, as Derand, De la Eue, Frezier, &c. But this method was never intended by them to be applied to Gothic rib vaulting. The Gothic style had grown out of use when De l'Orme wrote ; but as a matter of curiosity he has described the curves of a Gothic vault (in ch. viii.), in which every rib is perfectly independent of the other in its curvature. Each rib consists of a single arc of a circle whose centre is upon the impost level, and they cannot be there fore connected by projections. They all form pointed arches of different proportions, with the exception of the diagonal arch, which is very nearly a semicircle. The vaulting question consists of diagonal ribs, with one intermediate tierceron in each angle, as in fig. 10, but that the plan is a complete square. The ridge-rib or Heme e k is not horizontal, but slightly curved, so that the height of k above the impost plane is to that of e as about nine to eleven. The curve of the diagonal rib was first drawn (as De l'Orme himself says), then apparently the relative altitudes of the points e and f (that is, the lines e e p/) were determined, by drawing the proposed curve of the Heme ; and as the centres of the arcs are upon the level of the imposts a e a p, nothing remained but to draw the arcs of circles each through the two points t e l/. For, as I have already remarked, only two points are required to determine the circles when coupled with the condition that the centre shall be on the line of the impost. This may have been the genuine French Gothic method, but in our English examples the WILLIS ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE AGES 15 centres are commonly placed without respect to the impost level, and the general forms of the vault are different from those which are produced in this manner, as already mentioned. It is easy to see, however, that this method of De l'Orme is very far removed from the projection system, and that from its simplicity it was in all probability one of the rules of the olden time. Derand has also a chapter upon the Gothic vaults, but it appears to have been a mere comment upon De l'Orme, and indeed he wrote too late (1643) to be able to collect any genuine traditions of the Gothic methods ; whereas De l'Orme himself lived so close to the Gothic period * that he might well in his youth have been taught by Gothic masons. Derand, however, says, that in this style the ribs are always made arcs of circles, elliptical or other curves being inad missible, although he can see no reason why they should not be used, but that their effect upon the eye is not so agreeable (p. 177). He then proceeds to give the method of describing the arcs, which is manifestly a mere amplification of what De l'Orme had said. De la Eue is silent upon the subject, but Frezier, 1738, has given the same matter as Derand, amplified, and with many very sensible remarks appended thereto. f It is not, therefore, from the French writers that the absurd application of De l'Orme's projection method to Gothic ribs is due ; I believe this honour must be awarded to the ingenious Mr. William Halfpenny, { who, living at a period when Gothic architecture had sunk into com plete neglect, may very well be excused for having misapplied as he has done the projection system to the finding the " Mitre Arch of a Eegular Groin when the Intersecting Arches are Gothick ones " (p. 16). In this he has been followed by Mr. Nicholson,§ who has also taken much trouble to construct a Gothic vault with cuniconoidal surfaces and upon other fanciful hypotheses, which, as they produce curves for the ribs totally different from the genuine ones, can answer no purpose but that of destroying the mediaeval character of the work. || The south walk of the cloisters of Westminster has a vault, the ribs of which approach more closely in their appearance to projected ribs of this kind than any I have seen elsewhere, and I therefore took some pains to examine them, and found that they certainly were not so projected, but that the effect is due to the form of the middle plan, which is very nearly a square. Never theless the aspect of this vault is exceedingly flat and vapid compared with the bolder arrange ments of the vaults in the other walks of the cloister. In figs. 10 and 13 I have placed the elevation of each curve upon the plan of its own rib. This neat arrangement is employed by De l'Orme in many of his diagrams, and has been followed by all writers up to the present time. Of course, by adopting it, I do not mean to assert that it was followed by the Middle Age workmen, but rather I have employed it as a compendious and universally understood formula for setting out. The methods given in these figures would be the same if the curves were drawn separately by the side of the plan, which De l'Orme has actually done in his Gothic diagram, after the manner of fig. 11 or 16. It may be observed also that he has taken the pains also to draw moulds for the stones of the vaulting surfaces or pendentifs as he terms them. This, according to his own confession, was very seldom done, these vaults being more usually formed of brick or of stones so small or at least so thin that the curvatures and angles of their beds might be neglected, which indeed is plainly the case in all the existing English examples that I have examined. It must be clearly understood that I do not suppose the method of fig. 10 to be a restoration of an ancient practice. My object is to show clearly the mutual geometrical dependence upon each other of these three things which belong to a vault upon a given plan with single-arc ribs, namely, the form of the ridges, the middle plan of the spandrel, and the curvature of the ribs. Of these, if any two be * He flourished from 1536 to 1577. The contract for J Art of Sound Building, 1725. King's College vault is dated 1513, and the first stone of § Builder's Director, new edit. 1834, p. 79. St. Peter's at Rome was laid in 1506. || Treatise on Masonry and Stonecutting, 1828, p. 91. Traite de Stiriotomie, t. iii. p. 23, 3rd edit. 1769. 16 WILLIS ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE AGES determined, the other follows as a matter of course. It seems, therefore, most natural for us to select the two for determination which are the most easily appreciable by the eye, and these are the ridges and the middle plan. The method which I have given proceeds in this manner, and furnishes a general process by which we may imitate the various arrangements of these two things in the specimens of different ages. In the old time, one style alone was practised in each period, and a few simple rules were sufficient for the purpose. The change or improvement of one or more of these rules introduces new features and new characters, but which still are alone employed as long as they last, and until they are in turn superseded. But we, imitators of all styles, must have more comprehensive and flexible rules, capable of imparting to our works the characters of every age in turn. This necessarily gives to the methods which we invent and employ a much greater degree of com plication than is likely to have belonged to the rude practical geometry of the Middle Age work men ; each of our constructions being in fact a general formula, which includes many particular instances, every one applicable to a separate period. Notwithstanding, therefore, that I have employed the middle plan in determining the curvatures, I think it most likely that the different forms of the middle plans which may now be observed resulted from different rules for finding the centres and radii of the ribs, which were employed by the different schools of workmen according to their age or nation. In fact, in some examples the geometrical difficulties are very clumsily botched over, while in others the ribs are so beautifully adjusted in their relations that the greatest possible boldness and grandeur of effect is brought out, and to a degree which never appears after the four-centred arches are introduced. When the centres of single-arc ribs are placed upon the impost level, we have no longer the liberty of disposing at pleasure both of the form of the ridge and of the middle plan of the spandrel. One of these being settled, the other follows as a matter of course. It would lead me too far into geometrical investigations for the nature of the present paper were I to follow out these principles to their consequences. This I must leave for a future opportunity. But in this section my purpose is rather to direct investigation towards certain points and characteristic features in the adjustment of the ribs, which appear to me to exercise a great influence over the form of the vault, and even over its aesthetic character, and to the neglect of which may be attributed much of the feebleness of appearance in modern Gothic vaulting. In vaults of the Eoman and Italian styles, of which the groins are without ribs, the vaulting surfaee is the leading feature, and the disposition of it the only object to be attended to. But in Gothic ribbed vaults, on the contrary, the ribs are the principal features, and the surface of the vaults subordinate. To maintain this subordination oi the vaulting surface to the ribs, the latter should branch off from the abacus with the greatest possible appearance of mutual independence as separate arches, an appearance which is better given by single-arc ribs than by double-arc ribs or semi-four-centred arches. Also the vaulting surfaces or pannels of contiguous compartments should by no means have the appearance of continuity, which is given by the projection system, but which immediately suggests the idea that the surfaces really constitute the mechanical vault independently of the ribs, which seem to have been subsequently added, and might be removed without destroying the vault ; instead of which the ribs really support the vault, and should appear to do so in the decorative as well as in the mechanical construction. The apparent mutual independence of the ribs is increased in the best specimens by the manner in which they start from the abacus, some being more prominent than others. This is easily managed in the diagram fig. 10, by placing the feet of the curves at different distances from the centre A of the abacus. I have shown (to return from this digression) that a very simple geometrical process enables us to obtain the curvature and form of every rib from the two plans and the heights of the ribs WILLIS ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE AGES 17 FIG. 11. combined, but that without some such process they could not have been arranged with so manifest a power over the effect of the combination as the existing examples make evident. In vaults, however, whose ribs consist each of a single arc of the circle, every rib of the group will spring off from the common abacus at a different angle, if the centres of their respec tive circles be some above and some below the level of that abacus. In the first case the ribs will be slightly horseshoed or stilted, and in the second will start abruptly forward. These discrepancies may be seen in the majority of these vaults, and it was perhaps to remedy them that four-centred arches were introduced for the ribs, so that each rib should consist of two arcs of circles combined so as to have a common tangent at their junction. For when ribs are thus formed of two arcs, any number of them may start from the abacus at the same angle and even with the same curvature, and yet may have each a different height or span under certain simple limitations. Now a four-centred arch, or, which is the same thing, a two-centred rib, may have its upper radius adjusted to accommodate different spans and heights in two ways, when its lower radius remains constant. First, fig. 11, let a d c b be the quarter plan of a vault, and, for simplicity, suppose the ridge rib d b level, so that all the ribs will have the same height at the apex. Transferring the lengths of the ribs upon the plan to a e and a b, set up d h e g bf equal to each other and to the height of the ribs, and let the arc a e, whose centre is e, be the given lower circle of the ribs, which is to be the same in all. Produce e e to h, then any circle struck through e whose centre is upon this line will touch the circle a e at e, and thus answer the purpose of the upper circle of the rib. By taking different radii for this upper circle, therefore, we can make the ribs pass through the points hgf, as shown in the diagram, where h, g and p are the respective centres of the ribs e h e g e/. *In this example the ribs not only start all with the same curvature, but the change of curvature is made at the same height e in all, consequently the upper circles have all a different radius. Secondly, let it be required that the upper circles shall all have the same radius as well as the lower circles. In fig. 12, a d d h is the span and height of one rib, a b b/ those of another rib. c is the centre of the lower circle a b, which is common to both ribs. But as the upper circles are to be also the same, the curvature cannot change at the same height in both, b h being the given radius, the centre of one upper curve b h is at h, and of the other cf at p. It is easy to see that these centres will lie in the circumference of a circle whose centre c coincides with that of the lower circles and whose radius c h is the difference between the radii of the lower and upper circle of the ribs ; also a centre can be found at once, so that its arc shall pass through any given point, as /, by intersecting the locus h p with an arc whose centre is/ and radius that of the upper circles. Which of these principles of adjustment were employed by the Middle Age architects I do not know, but it would be very desirable to measure accurately a good number of examples, to ascertain whether the curvature of the upper circles is often the same in all the ribs as well as the lower. There are two working drawings of vaults, with intermediate four-centred ribs, in Pugin's * The centres are easily found in each case, as for e/ by joining b/, bisecting and drawing a perpendicular, which will intersect the line e h in the required centre r. FIG. 12. "3 Is ::;4" 18 WILLIS ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE AGES Examples, upon which the curvatures of the ribs are indicated, apparently with great accuracy. In one of them, the gateway of Magdalen College, the ridge ribs are horizontal, and consequently the arched ribs all of the same height. Each rib consists of two arcs of circles, but the lower circle is of very small diameter with respect to the upper circle, its radius being about one-ninth of the latter, and the same in all the ribs. All the upper circles of these ribs appear in this drawing to have the same radius, which is equal to the span of the transverse arch. The other example is the groining of a bay window of Eltham Palace, also having one inter mediate rib between each diagonal and wall rib. The ridge ribs are not horizontal, consequently in describing the curves of the arched ribs, the ridge ribs were probably first determined to give the heights of these ribs. Each rib, being half a four-centred arch, consists of two arcs of circles, the radius of the lower arc being a little less than half that of the upper arc, and this latter radius is again equal to the span of the smallest arch. These may be accidental proportions ; however, the two radii are respectively the same in every rib, and consequently the different heights and spans are accommodated, as in fig. 12, by employing different proportional lengths of the two circles in each rib. The centre of the lower circle is in all placed on the impost level, and this I believe to be universal in four-centred arches. The wall rib, in examples of this class, is often different both in its lower circle and upper circle from the other ribs, and in some cases the accommodation of heights is but clumsily effected. In the vault of Queens' College gateway at Cambridge, fig. 16, the ridge ribs are horizontal. The elevations of the principal ribs are shown upon g a, where G m is the curve of the wall rib G f, g n that of the transverse rib g a, and G q p that of the diagonal rib G k. The wall rib has a lower circle whose centre is t, and radius t k greater than those of the other ribs, and its upper circle k m is different, so that this rib is quite different from the others. The diagonal rib, transverse rib G a, and tiercerons have all the same lower circle G y with radius s y, and the same upper circle y n, so that the curvatures of these ribs are identical up to q, but the different spans are accommodated by breaking the curve off abruptly, as from q to p. As this change of direction takes place near the bosses b, c, d, it is scarcely perceptible unless closely looked after. The vaults of the side chapels of King's College, Cambridge, are excellent specimens ; those towards the west end, namely, four on the north side and three on the south side, have four fan-vaults similar to that of the chapel itself, and the two extreme north-eastern chapels have very elegant vaults of an earlier date than the great vault. The remainder of these small chapels are vaulted with plain vaults, which are included in the same contract with the fan vaults, and are specified as to be made of " a more coarse werke " (vide contract in Britton's Ant., vol. i.). These chapels are parallelograms of 20 feet 6 inches by 12 feet. The plain vaults have intermediate ribs or tiercerons, precisely in plan the same as fig. 9. The ribs are, however, two-centred and all of the same curvature, and the ridge ribs consequently have the figure of fig. 20. The vault is built of rib and pannel work. The vaults of the two north-eastern chapels are of the class which I have in a succeeding part of this paper denominated Heme vaults, and have an elegant pattern of the stellar kind, which is so contrived as to form not only stars round the angles, but also a six-rayed star round the centre. It has in all twenty-three bosses. This vault is also built of rib and pannel work, and the boss stones are formed on the principle of fig. 16, each stone including a portion of pannel between the stump of the ribs. The curvatures are beautifully managed. The ridges are both fig. ie. WILLIS ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE AGES 19 FIGJL. horizontal, and the ribs all leave the impost with the same curvature, and apparently their upper or crown curves have all the same radius, the different spans being accommodated on the principle of fig. 12. The transverse arch is very nearly equal in height to its own span. On the whole, these vaults are excellent specimens, but are seldom seen, as the chapels are shut up and neglected, and used as a workshop or lumber place. I am indebted to the kindness of Charles Barry, Esq., for the exact measures and curvatures of the vault of the crypt of St. Stephen's Chapel, at Westminster, which follow. The details and arrangements of this crypt have been published in various well-known works, and therefore the diagram plan of one quarter of a bay, in fig. A, will be sufficient for our present purpose, which is merely to give the curvatures of the ribs. a is the central boss of the vault. b c d e, the other bosses. ( a \ r^\ Q f, one of the piers. The respective diameters of the bosses are as follows : — Boss a, 3 feet 3 inches diameter. b, 1 „ 5 „ c, 1 „ 3 d, 1 „ 7 E, „ 10 The position of c on the plan is given by the perpendiculars c m c n, from the centre of the boss upon the lines a d d d respectively, of which c m = 3 feet 0| inches and c n = 3 feet 11 inches. The ribs a d a b and b e are all straight, but d c and a c are slightly curved ; the first having a rise of 1 inch and the second of | inch. The curvature of the ribs Aa Bb c c d d b e are shown in the following table, the explanation of which will be found on the succeeding page. These curvatures were measured by vertical ordinates which rise from a horizontal base at the level of the abacus or impost line of the pier p, and are for the most part taken at a dis tance of 1 foot apart, but in some places this equal spacing of the ordinates has been interfered with by local circumstances or obstruc tions. The table is arranged in this manner : m a, fig. B, being one of the ribs, let m a be the horizontal line or base, and a a the central ordinate, therefore a will be the crown of the arch or rib. b b c c d d, &c, are the other ordinates in sufficient number. Then the feet of these ordinates are indicated by the capital letters in order, beginning with a for the central one ; and the upper extremity of each ordinate is indicated by the small letter which corresponds to the capital at its foot. In this way, a a, b b, c c, &c, will indicate the height of the ordinates, and ab, b c, c d, &c, their respective horizontal distances. The ordinate at the springing, as m, will be necessarily equal to zero. By following this system, a great number of curvatures may be arranged and published without requiring engraved figures. I have laid down these ordinates upon a large scale, with a view to determine the radii of the circles. However, it will always be found in this operation that the irregularity of the workmanship and the settlement of the work will occasion such deviations from the original form, as to leave the exact radius and centre of the arc a matter of some doubt. For this reason, the fairest and best plan is to publish the ordinates in every case, without attempting to form FIG.B. 20 WILLIS ON THE CONSTRUCTION OF THE VAULTS OF THE MIDDLE AGES hypotheses ; for it will only be by collecting a great number of such examples, and comparmg them, that we can hope to deduce general rules. I will merely state the radii as far as I have been able to determine them. Curvatures of the Bibs in the Crypt of St. Stephen's Chapel, in feet and inches. "3 » ¦Si O